Experiments with cellular structures

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The objects of this proposal are five classes of algebras of interest in representation theory, knot theory and mathematical physics:– Classical Schur algebras (associated with the algebraic group GLn),– group algebras of symmetric groups,– Brauer algebras,– partition algebras,– the recently discovered Schur algebras of Brauer algebras.We will focus on a crucial and computationally accessible part of their cellular structure, the multiplication inside cellular subquotients, and on the information about simple modules, semisimplicity and other fundamental properties to be derived from this multiplication. The project consists of three main parts: First, we will develop algorithms and computer programs to calculate the relevant structure constants, even for large examples. Secondly, we will run experiments, especially by varying some of the discrete and of the continuous parameters of the algebras. Thirdly, we will use the output of the experiments to formulate precise conjectures about structure and independence of parameters and to get starting points for proofs of general results.